NCERT Class 12-Mathematics: Chapter –1 Relations and Functions Part 6

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Question 4:

Let be the function defined by . write .

Answer:

Question 5:

If and the function , write .

Answer:

Question 6:

If is defined by , write .

Answer:

Question 7:

Is a function? If is described by , then what value should be assigned to and .

Answer:

Question 8:

Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.

(i) .

(ii).

Answer:

(i) Represents function which is surjective but not injective

(ii) Does not represent function.

Question 9:

If the mappings f and g are given by

and , write .

Answer:

Question 10:

Let C be the set of complex numbers. Prove that the mapping given by , is neither one-one nor onto.

Answer:

We have,

given by

In order to prove that f is one-one, it is sufficient to prove that, .

Let and are two distinct complex numbers.

Now,

Here, we observe that

This shows that different element of C may have the same value in R.

Thus, is not one-one.

is onto if every element of R is the f-image of some element of C.

We have, and

We observe that negative real numbers in R do not have their pre-images in C.

Thus, is not onto.

Hence, is neither one-one nor onto.

Question 11:

Let the function be defined by . Show that is neither one-one nor onto.

Answer:

We have,

In order to prove that f is one-one, it is sufficient to prove that , .

Let and are two different elements in R.

Now,

We observe that .

This shows that different element in R may have same image.

Thus, is not one-one.

We know that lies between and .

So, the range of f is which is not equal to its co-domain.

i.e., range of

In other words, range of is less than co-domain, i.e. there are elements in co-domain which does not have any pre-image in domain.

So, f is not onto.

Hence, is neither one-one nor onto.